Comprehensive coverage of the prior art may be found, for example, in W. K. Pratt, “Digital Image Processing,” 2nd Edition, John Wiley and Sons, NY (1988); H. Stark, “Image Recovery; Theory and Application,” Academic Press, Inc., Harcout Brace Jovanovich Publishers, New York (1987); R. C. Gonzalez and P. Wintz, “Digital Image Processing,” 2nd Edition, Addison-Wesley Publishing Company, Inc., Advanced Book Program, Reading, Mass. (1987); and R. L. Lagendijk and J. Biemond, “Iterative Identification and Restoration of Images,” Kluwer International Series in Engineering and Computer Science, Kluwer Academic Publishers, Boston, Mass., (1991).
There have been several attempts to develop a method of image processing and restoration based on the solution of the linear ill-posed inverse problem:d=Bm,  (1) where d is the blurred (or degraded) image, m is original (or ideal) image, and B is the blurring linear operator of the imaging system. Note that the original image, as well as the blurred image, can be defined in a plane (2-D image: m=m(x,y), d=d(x,y)) or in a volume (3-D image: m=m(x,y,z), d=d(x,y,z)).
A wide variety of electron-optical devices obey equation (1) with different blurring operators as noted by C. B. Johnson et al., in “High-Resolution Microchannel Plate Image Tube Development,” Electron Image Tubes and Image Intensifiers II, Proceedings of the Society of Photo-Optical Instrumentation Engineers, Vol. 1449. I. P. Csorba, Ed. (1991), pp. 2-12. These devices are used in various biomedical imaging apparatus, including image intensifier-video camera (II-TV) fluoroscopic systems (see S. Rudin et al., “Improving Fluoroscopic Image Quality with Continuously Variable Zoom Magnification,” Medical Physics. Vol. 19 (1991), pp. 972-977); radiographic film digitizers (see F. F. Yin et al., “Measurement of the Presampling Transfer Function of Film Digitizers Using a Curve Fitting Technique,” Medical Physics, Vol. 17 (1990), pp. 962-966); radiographic selenium imaging plates (see P. J. Papin and H. K. Huang, “A Prototype Amorphous Selenium Imaging Plate System for Digital Radiography,” Medical Physics, Vol. 14 (1987), pp. 322-329); computed radiography systems (see S. Sanada et al., “Comparison of Imaging Properties of a Computed Radiography System and Screen-Film Systems.” Medical Physics, Vol. 18 (1991), pp. 414-420; H. Fujita et al., “A Simple Method for Determining the Modulation Transfer Function in Digital Radiography,” IEEE Transactions on Medical Imaging, Vol. 11 (1992), pp. 34-39); digital medical tomography systems (see M. Takahashi et al., “Digital 'IV Tomography: Description and Physical Assessment,” Medical Physics, Vol. 17 (1990), pp. 681-685).
Geophysical, airborne, remote sensing, and astronomical blurred images also can be described by equation (1) (see M. Bath, “Modern Spectral Analysis with Geophysical Applications,” Society of Exploration Geophysicists, (1995), 530 pp.; C. A. Legg, “Remote sensing and geographic information systems,” John Wiley & Sons, Chichester, (1994), 157 pp.).
Most prior efforts to solve the problem: (1) are based on the methods of linear inverse problem solutions. Inverse problem (1) is usually ill-posed, i.e., the solution can be non-unique and unstable. The conventional way of solving ill-posed inverse problems, according to regularization theory (A. N., Tikhonov, and V. Y., Arsenin, “Solution of ill-posed problems,” V. H. Winston and Sons., (1977); M. S., Zhdanov, “Tutorial: regularization in inversion theory,” Colorado School of Mines (1993)), is based on minimization of the Tikhonov parametric functional:Pα(m)=φ(m)+αs(m),  (2) where φ(m) is a misfit functional determined as a norm of the difference between observed and predicted (theoretical) degraded images:φ(m)=∥Bm−d∥2=(Bm−d, Bm−d).  (3) Functional s(m) is a stabilizing functional (a stabilizer).
It is known in the prior art that there are several common choices for stabilizers. One is based on the least squares criterion, or, in other words, on L2 norm for functions describing the image:SL2(m)=∥m∥2=(m,m)=∫vm2dv=min,   (4) where V is the domain (in 2-D space or in 3-D space) of image definition, and ( . . . , . . . ) denotes the inner product operation.
The conventional argument in support of this norm comes from statistics and is based on an assumption that the least square image is the best over the entire ensemble of all possible images.
Another stabilizer uses minimum norm of difference between the selected image and some a priori image mapr:SL2apr(m)=∥m−mapr∥2=min.  (5) This criterion, as applied to the gradient of image parameters ∇m, brings us to a maximum smoothness stabilizing functional:smax sm(m)=∥∇m∥2=(∇m, ∇m)=min.   (6) Such a functional is usually used in inversion schemes. This stabilizer produces smooth images. However, in many practical situations the resulting images don't describe properly the original (ideal) image. Inversion schemes incorporating the aformentioned functional also can result in spurious oscillations when m is discontinuous.
It should be noted that in the context of this disclosure, “image parameters” is intended to describe the physical properties of an examined media. Such parameters include without limitation: color, intensity, density, field strength, and temperature. Discrete samples of such parameters are assembled, eg. in a matrix [d], to describe an image of the media. In the paper by (L. I., Rudin, Osher, S. and Fatemi, E., “Nonlinear total variation based noise removal algorithms,” Physica D, 60, (1992), pp. 259-268) a total variation (TV) based approach to reconstruction of noisy, blurred images has been introduced. This approach uses a total variation stabilizing functional, which is essentially L1 norm of the gradient:sTV(m)=∥∇m∥L1=∫v|∇m|dv.  (7) This criterion requires that an image's parameter distribution in some domain V be of bounded variation (for definition and background see (E., Giusti, “Minimal surfaces and functions of bounded variations,” Birkhauser (1984)). However, this functional is not differentiable at zero. To avoid this difficulty, Acar and Vogel introduced a modified TV stabilizing functional (R., Acar, and C. R., Vogel, “Analysis of total variation penalty methods,” Inverse Problems, 10, (1994), pp. 1217-1229):sβTV(m)=∫v√{square root over (|∇m|2+e2)}dv.  (8) The advantage of this functional is that it doesn't require that the function m is continuous, but just that it is piecewise smooth (C. R., Vogel, and M. E., Oman, “A fast, robust algorithm for total variation based reconstruction of noisy, blurred images,” IEEE Transactions on Image Processing, (1997)). The TV norm doesn't penalize discontinuity in the image parameters, so we can remove oscillations while sharp conductivity contrasts will be preserved. At the same time it imposes a limit on the total variation of m and on the combined arc length of the curves along which m is discontinuous.
TV functionals sTV(m) and sβTV(m), however, tend to decrease bounds of the image parameters variation, as can be seen from (7) and (8), and in this way still try to “smooth” the real image, but this “smoothness” is much weaker than in the case of traditional stabilizers (6) and (5).
In yet another development A. S. Carasso introduced a procedure for digital image restoration based on Tikhonov regularization theory and description of the blurred image d as the end result of a diffusion process applied to the desired ideal image m (A. S., Carasso, “Overcoming Holder Continuity in Ill-Posed Continuation Problems,” SIAM Journal on Numerical Analysis, Volume 31, No. 6, December 1994, pp. 1535-1557; C. E., Carasso, U.S. Pat. No. 5,627,918, May 1997). In the framework of Carasso's method the image restoration procedure is equivalent to solving diffusion equation backwards in time using the given degraded image d as data at the final moment of diffusion. The desired ideal image is the corresponding solution of the diffusion equation at the initial time moment. To generate a stable solution, Carasso suggests to use a constraint∥m−B5m∥=min, where B5 is the fractional power of the operator B.
The main limitation of Carasso's approach is that it can be applied only to a specific class of integral blurring operators, which can be described by the diffusion problem solution.
In yet another development (H. Kwan, and J. T., Liang, Digital image sharpening method using SVD block transform, U.S. Pat. No. 4,945,502, Jul. 1990), the authors suggest to use the Singular Value Decomposition technique and a priori known information about the statistical properties of the noise and image texture to suppress the noise in the degraded images.
The foregoing attempts met with varying degrees of success in developing the methods of digital image restoration. However, from the point of view of practical applications, these methods are not entirely satisfactory. Therefore, it is highly desirable to develop a method which can be applied to a wide variety of image restoration problems.